| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question requiring binomial probability calculations and critical region determination. Part (a) involves routine cumulative probability lookups from tables, while part (b) is a one-tailed test with clearly stated hypotheses. Both parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(X \sim B(20, 0.25)\) | M1 | May be implied by correct CR |
| \(P(X \geq 10) = 1 - 0.9861 = 0.0139\) | A1 | awrt 0.0139 |
| \(P(X \leq 1) = 0.0243\) | A1 | awrt 0.0243 |
| \((0 \leq)X \leq 1 \cup 10 \leq X(\leq 20)\) | A1A1 | \(X \leq 1\) or \(0 \leq X \leq 1\) or \([0,1]\); \(X \geq 10\) or \(10 \leq X \leq 20\) or \([10,20]\). NB: CR written as \(1 \geq X \geq 10\) gets A1 A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.25\), \(H_1: p < 0.25\) | B1 | Both hypotheses with \(p\) |
| \(X \sim B(20, 0.25)\) | ||
| \(P(X \leq 3) = 0.2252\) or CR \(X \leq 1\) | M1A1 | Using \(B(20,0.25)\); awrt 0.2252 (allow 0.7748 if not using CR) |
| Insufficient evidence to reject \(H_0\); not significant; 3 does not lie in CR | M1d | Correct statement (dependent on previous M) |
| No evidence that the changes to the process have reduced the percentage of defective articles | A1cso | Must contain: changes/new process, reduced, number/percentage, defective articles. No incorrect working. |
# Question 6:
## Part 6(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $X \sim B(20, 0.25)$ | M1 | May be implied by correct CR |
| $P(X \geq 10) = 1 - 0.9861 = 0.0139$ | A1 | awrt 0.0139 |
| $P(X \leq 1) = 0.0243$ | A1 | awrt 0.0243 |
| $(0 \leq)X \leq 1 \cup 10 \leq X(\leq 20)$ | A1A1 | $X \leq 1$ or $0 \leq X \leq 1$ or $[0,1]$; $X \geq 10$ or $10 \leq X \leq 20$ or $[10,20]$. NB: CR written as $1 \geq X \geq 10$ gets A1 A0 |
## Part 6(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.25$, $H_1: p < 0.25$ | B1 | Both hypotheses with $p$ |
| $X \sim B(20, 0.25)$ | | |
| $P(X \leq 3) = 0.2252$ or CR $X \leq 1$ | M1A1 | Using $B(20,0.25)$; awrt 0.2252 (allow 0.7748 if not using CR) |
| Insufficient evidence to reject $H_0$; not significant; 3 does not lie in CR | M1d | Correct statement (dependent on previous M) |
| No evidence that the **changes** to the process have **reduced** the **percentage** of **defective articles** | A1cso | Must contain: changes/new process, reduced, number/percentage, defective articles. No incorrect working. |
---6. In a manufacturing process $25 \%$ of articles are thought to be defective. Articles are produced in batches of 20
\begin{enumerate}[label=(\alph*)]
\item A batch is selected at random. Using a $5 \%$ significance level, find the critical region for a two tailed test that the probability of an article chosen at random being defective is 0.25\\
You should state the probability in each tail which should be as close as possible to 0.025
The manufacturer changes the production process to try to reduce the number of defective articles. She then chooses a batch at random and discovers there are 3 defective articles.
\item Test at the $5 \%$ level of significance whether or not there is evidence that the changes to the process have reduced the percentage of defective articles. State your hypotheses clearly.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2013 Q6 [10]}}